Successive numbers starting from $1$ $1$ are written in a rectangular grid, starting in the top left corner and snaking down diagonally to the right as shown. In which row and column does $2008$ $2008$ occur?

Question

Successive numbers starting from $1$ $1$ are written in a rectangular grid, starting in the top left corner and snaking down diagonally to the right as shown. In which row and column does $2008$ $2008$ occur?

$01002006007015016$ $color(white)(0)1color(white)(00)2color(white)(00)6color(white)(00)7color(white)(0)15color(white)(0)16$
$030050080140 etc$ $color(white)(0)3color(white)(00)5color(white)(00)8color(white)(0)14color(white)(0)"etc"$
$04009013$ $color(white)(0)4color(white)(00)9color(white)(0)13$
$10012$ $10color(white)(0)12$
$11$ $11$

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Answer 1 · 2017-02-06T07:28:04

The $9$ $9$ th row, $55$ $55$ th column.

Explanation:

The last number added to the $n$ $n$ th (reverse) diagonal is the $n$ $n$ th triangular number $T_{n} = \frac{1}{2} n (n + 1)$ $T_n = 1/2n(n+1)$ .

What is the smallest triangular number greater than or equal to $2008$ $2008$ ?

Given:

$\frac{1}{2} n (n + 1) = T_{n} \geq 2008$ $1/2n(n+1) = T_n >= 2008$

Multiply both ends by $2$ $2$ to get:

$n (n + 1) \geq 4016$ $n(n+1) >= 4016$

So the value of $n$ $n$ we are looking for is roughly $\sqrt{4016} \approx 63$ $sqrt(4016) ~~ 63$ and we find:

$T_{63} = \frac{1}{2} \cdot 63 \cdot 64 = 2016$ $T_63 = 1/2*63*64 = 2016$

$T_{62} = \frac{1}{2} \cdot 62 \cdot 63 = 1953$ $T_62 = 1/2*62*63 = 1953$

So $2008$ $2008$ lies on the reverse diagonal which has $T_{63} = 2016$ $T_63 = 2016$ at one end.

Note that due to the boustrophedonic (like an ox ploughing a field) way in which the numbers are written, $T_{63}$ $T_63$ will be written on the top row as the $63$ $63$ rd term of that row.

$2008$ $2008$ will occur $8$ $8$ rows below and to the left of it, i.e. on the $9$ $9$ th row in the $55$ $55$ th column.

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Successive numbers starting from $1$ $1$ are written in a rectangular grid, starting in the top left corner and snaking down diagonally to the right as shown. In which row and column does $2008$ $2008$ occur?

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Explanation:

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Successive numbers starting from 11 are written in a rectangular grid, starting in the top left corner and snaking down diagonally to the right as shown. In which row and column does 20082008 occur?

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Successive numbers starting from $1$ $1$ are written in a rectangular grid, starting in the top left corner and snaking down diagonally to the right as shown. In which row and column does $2008$ $2008$ occur?